The present invention provides: (1) The exact characteristic equation of vibrations of a circular cylindrical shell; (2) The characteristic equation is good for any linear viscoelastic materials including elastic solids and non-Newtonian fluids; (3) Computer programs to generate data for the vibrational wave spectrum analysis; (4) The physical and geometrical dimensions of the shafts can be determined between two critical frequencies.
High speed rotating shafts are one of the most important parts of machines that are used for the transformation of other energies into kinetic energy. They are used in turbines, rotors, motors, pumps and flywheels that are the most vital components of space shuttles, jetliners, and all other vehicles operated in space, air, land and in hydrospace. Thus far, the design of low and moderate speed rotating shafts has been very well known and successfully applied and used in our daily machines. However, because of the need for higher performance of the machines in our modem times, special care must be given to design high speed rotating shafts. The mechanical structural failure of the shaft may lead to a disaster of losing human lives in airplanes because fragments of the broken shaft carry high kinetic energy that could destroy airplane components in their paths. If this is allowed to happen, the airplane will be shattered into pieces.
To solve the problem, two aspects have been followed: one by means of experimental tests; the other by means of theoretical analysis. They are complimentary to each other. The former provides a practical solution of the problem. It is more expensive than using the latter on account of the choices of physical and geometrical parameters of various modem materials of the shafts for different purposes. The latter provides a systematic guide for the tests to choose the appropriate material for the shaft once the analysis is completed and perfected with a fully developed computing program.
From the theoretical aspect, the proposed solution of the problem has been attempted by means of three methods: 1. particle dynamics; 2. structural dynamics; and 3. elastodynamics. Historically, the particle dynamics approach to solve the rotary dynamic problems can be traced back to two papers in 1895 by Dunkerly, S., xe2x80x9cOn the Whirling and Vibration of Shafts,xe2x80x9d Phil. Trans. Roy Soc. A., Vol. 185, pp. 269-360, and by Foppl, A., xe2x80x9cDas Problem der Laval""schen Turbinewlle,xe2x80x9d Civilinggenieur 41, pp. 332-342. The interests of those problems have been extended and continued by many others in the academic and industrial organizations. Most recently, the paper by Crandall, S. H., xe2x80x9cThe Physical Nature of Rotor Instability Mechanismsxe2x80x9d in xe2x80x9cRotor Dynamical Instabilityxe2x80x9d (M. L. Adams, ed.) ASME Special Publication, AMD-Vol. 55, pp. 1-18 (1983), provided simple physical explanations of several instability mechanisms. They are whirling due to Coriolis acceleration; internal damping in the rotor; flow about the rotor; and internal flow within a rotor. It is well known that the dynamics of particles approach involves the vibration of masses between springs. Similarly, the vibration of continuous structures are also usual topics in the field of structural mechanics. A combination of both, and the efforts of many as indicated by the many papers by Myklestad, N.O., xe2x80x9cA New Method of Calculating Natural Modes of Uncoupled Bending Vibration of Airplane Wings and Other Types of Beamsxe2x80x9d J. Aero. Sci., pp.153-162 (April 1944), Prohl, M. A., xe2x80x9cA General Method for Calculating Critical Speeds of Flexible Rotors.xe2x80x9d Trans. ASME A-142 (September 1945), Pestel, E. C., and Leckie, F. A., xe2x80x9cMatrix Methods in Elastodynamicsxe2x80x9d McGraw-Hill, N.Y. (1963) and by Thomson, W. T., xe2x80x9cMatrix Solution for the Vibration of Nonuniform Beams.xe2x80x9d J. Appl""d Mech., pp.337-339 (September 1950) and xe2x80x9cVibration Theory and Application,xe2x80x9d Prentice-Hall, Englewood Cliff, N.J. (1965), brings about a method to solve many complicated structural problems.
The most important element of the structural dynamics approach is the transfer matrix of the structural system. As can be seen from the books by Thomson and by Pestel and Leckie, the transfer matrix can be derived from the governing differential equations of a physical problem. A great deal of these have been done for the beam theories on torsional, axially compressional and transversal vibrations. However, the transfer matrix for the exact theory of elastodynamics and thermoelastodynamics is comparatively unknown. This was so because general solutions of the governing equations of elastodynamics were either too tedious or uninformative for any practical uses. One of these examples is the well known Pochhammer and Love solution of the flexural vibration of an elastic rod. See, Pochhammer, L., xe2x80x9cUeber die Fortflanzungsgeschwindigkeiten Schwinggungen in ein Unbegrenzten Isotropen Kreiszylinderxe2x80x9d J. Fur Math., Vol. 81, pp. 324-336 (1876), and Love A. E. H., xe2x80x9cA Treatise on the Mathematical Theory of Elasticityxe2x80x9d Cambridge University Press, Fourth Edition, pp. 287-292 (1927). The solution was originated by Pochhammer in 1876 and made independently by Chree in 1886. The original Pochhammer-Chree solution of the problem was very tedious. Concise and modem systematic solution of the same problem together with the wave spectra analysis were separately provided by Gazis in 1959, Gazis, D. C., xe2x80x9cThree-Dimensional Investigation of the Propagation of Waves in Hollow Circular Cylinders I: Analytical Foundation, and II: Numerical Results, J. Ac. Soc. Amer., Vol 31, pp. 568-578, by Greenspon in 1960, Greenspon, J. E., xe2x80x9cVibration of a Thick-walled Cylindrical Shellxe2x80x94Comparison of the Exact Theory with Approximate Theories.xe2x80x9d, J. Ac. Soc. Amer., Vol 32, pp. 571-578, and by Wong in 1967, Wong, P. K., xe2x80x9cOn the Unified General Solution of Linear Wave Motions of Thermoelastodynamics and Hydrodynamics with Practical Examplesxe2x80x9d Transaction of ASME, Journal of Applied Mechanics, Vol 34, pp. 879-887 (December 1967) and Vol. 35, pp 847 (December 1968). The wave spectra were extended for the entire class of linear viscoelastic materials for solids and shells by Wong in 1970, Wong, P. K., xe2x80x9cWaves in Viscous Fluids, Elastic Solids, and Viscoelastic Materialsxe2x80x9d Ph.D. Dissertation, Department of Aeronautics and Astronautics, Stanford University, Stanford, Calif. (1970).
The invention is based upon elastodynamnic methods; and certain details can be traced from Wong, P. K., xe2x80x9cOn the Unified General Solution of Linear Wave Motions of Thermoelastodynamics and Hydrodynamics with Practical Examples.xe2x80x9d Transaction of ASME, Journal of Applied Mechanics, Vol 34, pp. 879-887 (December 1967) and Vol. 35, pp 847 (December 1968).
The invention illustrates two main features different from other approaches to solve the design of high speed rotating shafts: (1) since the general solutions of the governing equations of elastodynamics and thermoelastodynamics are shown by Wong in 1967, 1968 and 1970, the derivation of transfer matrices for the exact theories is therefore possible; and (2) it can be shown that the solutions are also useful for practical design purposes. These can be demonstrated in a practical example which can be solved both from the lumped mass technique and from the elastodynamics theory.
It is known that the lumped mass technique is a combination of particle dynamics and structural mechanics approaches. The comparison of lumped mass techniques with elastodynamics is discussed below and in connection with accompanying FIGS. 1-3. Consider, for example, an elastic solid rotating shaft of mass m, density xcex4, Young""s modulus E and area moment of inertia I=xcfx80r4/4. The shaft is simply supported by two bearings as shown in FIG. 1 for a homogeneous circular cylindrical rotating rod and its equivalence being replaced by a massless shaft with its equivalent mass m=xcfx80r2lxcex4 concentrating at the center of the rod as shown in FIG. 2. The natural frequency of transversal vibration of the system in FIG. No. 2 can be obtained from the vibrational theory of lumped massed technique:                               ω          n                =                                            k              m                                =                                    48              ⁢                              xe2x80x83                            ⁢                              EI                /                m                            ⁢                              xe2x80x83                            ⁢                              l                3                                                                        (        1        )            
Now let the mass m be that of the shaft itself, the natural frequency of transversal vibration of a rotating homogeneous circular rod shown by FIG. 1 can be expressed as                               ω          n                =                                            48              ⁢                              xe2x80x83                            ⁢                              EI                /                π                            ⁢                              xe2x80x83                            ⁢                              r                2                            ⁢              δ              ⁢                              xe2x80x83                            ⁢                              l                4                                              =                      2            ⁢                          3                        ⁢                          (                              r                                  l                  2                                            )                        ⁢                                          E                δ                                                                        (        2        )            
To compare the result from equation (2) with the natural frequencies obtained from the exact theory of elastodynamics for a circular rod in transversal vibration, one shall find that this is possibly close to the first natural frequency. The question is what about the second and the subsequent higher critical frequencies that are obtained from the exact theory. In order to obtain both the first and the second critical frequencies of the system, one may have to consider the problem represented as shown in FIG. 3. Obviously, in order to get the third frequency, one must divide the shafts into three equal masses each of which is m/3 at an equal interval of l/4 apart. Continuing this process as many times as possible, the ultimate result will be close to those from the exact theory. One may observe that the increase of the number of redistributed masses m will increase the number of critical frequencies. The characteristic equation (critical frequency equation or natural frequency equation of the system) obtained from this approach is a high degree algebraic equation, while the characteristic equation obtained from the exact theory for the same system is a transcendental equation.
Based upon the above reasoning, the accuracy of the lumped mass technique compared with the exact theory to obtain the critical frequencies for a vibrational systems depends upon the number of masses input and also upon how much resemblance of the real system, being replaced by springs and beams, to the remodeled system. Many rotary dynamics computing programs being used in industries were written from the principle of transfer matrix for beam theory including the effects of shear deflection and rotary inertia and with masses and springs attached. They are good for calculating the low critical frequencies. Their advantages over using the exact theory of elastodynamics are: (1) Those computer programs can be applied for analysis of nonuniform beams and shells, whereas the exact theory can be applied to uniform cylinders and shells only before the transfer matrix of the exact theory is fully developed; and (2) Even if the transfer matrix method of the exact theory is fully developed to handle beams of nonuniform cross-sections, the computation is still easier for the approximate theory because the former involves transcendental functions in the characteristic equation, while the latter, involves only algebraic functions in the characteristic equation. The disadvantage is that the critical frequencies obtained by theories other than the exact elastodynamics theory could be far from the exact values. Thus, concrete, systematic and precise wave spectra cannot be concluded so reliable as compared to the results from the exact theory. The wave spectra formed by connecting the critical frequencies of the entire vibrating system is the foundation to set the criterion for the design of rotating shafts. This leads to the need to find the characteristic equation for a vibrating shaft from the elastodynamics theory.
The invention provides computer programs and data for the design of high speed rotating shafts of elastic materials. The invention provides computer programs and data for the design of both high and low speed rotating shafts of linear viscoelastic materials. The invention provides computer programs and data for the design of fuel pins and the structure of the nuclear reactor cores. The invention provides computer programs and data for the design of composite structures in aerospace vehicles, hydro space vehicles and geophysical structures. The invention provides computer programs and data for the design of wide-band frequency spherical and circular cylindrical antennae. The invention provides computer programs and data for the design of hydraulic structural systems. The invention provides computer programs and data for the design of geo-thermal-mechanical structural systems. The invention provides computer programs and data for the design of the structures of coaxial cables. The invention provides computer programs and data for the design of biomechanical structural systems. The invention provides the general solutions of linear wave motions of the entire class of linear viscoelastic materials in viscoelastodynamics, thermoelastodynamics and hydrodynamics.